Over the past several decades there has been growing interest in the development of devices based on second-order nonlinear effects such as sum-frequency generation (SFG) and difference frequency generation (DFG) and devices based on third-order nonlinear effects such as Raman-resonant four-wave-mixing (FWM) and Kerr-induced four-wave-mixing (FWM). SFG, DFG and Kerr-induced FWM are parametric light-matter interactions that are not resonant with a material level and that are used in parametric converters and parametric amplifiers. Raman-resonant FWM is a light-matter interaction that is perfectly resonant or almost perfectly resonant with a characteristic energy level of the material such as a vibrational energy level and that is used in Raman converters. SFG, DFG and Kerr-induced FWM involve a pump radiation beam at frequency ωp, a signal radiation beam at frequency ωs, and an idler radiation beam at frequency ωi. Raman-resonant FWM involves a pump radiation beam at frequency ωp, a Stokes radiation beam at frequency ωstrokes that is lower than the pump frequency, and an anti-Stokes radiation beam at frequency ωanti-stokes that is higher than the pump frequency. One also uses the terms signal and idler for the Stokes and anti-Stokes radiation beams, respectively, or vice versa, and uses ωs and ωi to denote their frequencies. Due to the wavelength versatility offered by SFG, DFG, Raman-resonant FWM and Kerr-induced FWM, these processes feature a multitude of application possibilities in different domains such as optical communication and spectroscopy.
Basically, Raman-resonant FWM and Kerr-induced FWM are interactions between two pump photons, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωi satisfy the relation 2ωp−ωs−ωi=0. For Raman-resonant FWM one has in addition that |ωp−ωs|=2π×ΔR, with ΔR being the Raman shift of the considered Raman-active material. In the case of SFG and DFG there is an interaction between 1 pump photon, one signal photon and one idler photon, and the frequencies of these photons ωp, ωs and ωi satisfy the relation ωp+ωs=ωi fpr SFG and ωp−ωs=ωi for DFG. The efficiency of all processes depends on the pump intensity and on the processes' phase mismatch. The linear part Δklinear of the phase mismatch for Raman-resonant FWM and Kerr-induced FWM is given byΔklinear=2kp−ks−ki where k{p,s,a}=ω{p,s,a}xn{p,s,a}/c are wave numbers with n{p,s,a} representing the effective indices of the pump, signal and idler waves, respectively. One can also write Δklinear as
      Δ    ⁢                  ⁢          k      linear        =            -                                    β            2                    ⁡                      (            Δω            )                          2              -                  1        12            ⁢                                    β            4                    ⁡                      (            Δω            )                          4            where β2=d2k/dω2 is the group velocity dispersion (GVD) at the pump wavelength, β4=d4k/dω4 is the fourth-order dispersion at the pump wavelength, and Δω the frequency difference between the pump and signal waves. For SFG the linear part Δklinear of the phase mismatch is given byΔklinear=kp+ks−ki 
For DFG the linear part Δklinear of the phase mismatch is given byΔklinear=kp−ks−ki 
The total phase mismatch for these processes also contains a nonlinear part that is function of the pump intensity, but since linear phase mismatches are considered here that are mostly much larger than the nonlinear part of the phase mismatch, the latter can be neglected in the remaining part of this text.
Due to their nonlinear nature, all above-mentioned processes perform best at high optical intensities. These can be obtained by tightly confining the light for example in a nanowire waveguide and also by employing ring structures, whispering-gallery-mode disk resonators, or any other resonator structure in which the incoming light waves are resonantly enhanced. Regarding the requirement of having a small effective phase mismatch for the wavelength conversion processes, much progress has been made over the past several years, in particular for converters based on silicon waveguides. For these converters, by engineering the dispersion of a silicon nanowire waveguide one can obtain phase-matched Kerr-induced FWM in the near-infrared for pump-signal frequency shifts with an upper limit of 52 THz (i.e., pump-signal wavelength differences up to 418 nm in the near-infrared region).
Notwithstanding the broad applicability of this phase-matched conversion technique, there are circumstances, applications, and materials where an alternative approach can be useful. First of all, not all materials used for SFG, DFG, Raman-resonant or Kerr-induced FWM are as easily workable as silicon to fabricate waveguide structures, which implies that not all materials can benefit from the waveguide-based phase-matching technique outlined above. Furthermore, even if one considers only a material such as silicon for which the waveguide-based phase-matching technique described above is well developed, it is important to know that, although the phase-matching bandwidth of the silicon nanowire referred to above is more than wide enough to enable phase-matched Raman-resonant FWM in the near-infrared at a pump-signal frequency shift of 15.6 THz, the dispersion-engineered geometry of the waveguide is such that does not comply with the fabrication constraints of multiproject-wafer-oriented silicon photonics foundries, which rely on conventional semiconductor processing technology and employ a standard waveguide thickness of typically 220 nm. Since the use of such foundries can pave the way to the large-volume fabrication of integrated photonic components at low cost, adhering to these foundry standards is a natural strategy to exploit the full potential of silicon photonic devices. Furthermore, the developed silicon-based converters often are not compact and require propagation distances of at least 1 cm to achieve substantial conversion efficiencies. This leads to device footprints that are too large for realizing cost-effective photonic integrated circuits. Also for SFG and DFG in silicon nanowires, there are dispersion-engineering issues. To establish SFG and DFG in silicon, one usually applies strain on a silicon nanowire to induce the second-order nonlinearity that is needed for these processes and hence make the nanowire a quadratically nonlinear optical medium. Because of the very large pump-signal frequency shifts typically used in SFG and DFG, it is practically impossible to engineer the dispersion of the strained silicon nanowires in such a way that phase-matched SFG or DFG is obtained. Hence, it is challenging to achieve efficient SFG or DFG using only dispersion engineering.
One suggestion has been to establish quasi-phase-matching for SFG, DFG, Raman-resonant FWM or Kerr-induced FWM by periodically modifying the material properties within the medium through which the light propagates. This traditional quasi-phase-matching technique for these nonlinear processes can be understood as follows: In case nothing is done about the phase mismatch, the idler intensity for radiation would continuously oscillate along the propagation path between a maximal value and zero, as the phase-mismatch-induced dephasing of the fields—this dephasing evolves periodically with the propagation distance—causes the nonlinear optical processes to either increase or decrease the idler intensity along the propagation path. When using traditional quasi-phase-matching for these processes, one adjusts the propagation regions behind the positions of maximal idler intensity, so that one does not have a total drop down of the idler intensity in these regions but at the same time the fields' dephasing, accumulated up to the positions of maximal idler intensity, can evolve back to zero in these adjusted regions. Hence, after traversing these adjusted areas the idler intensity can start growing again towards a maximum. The type of “adjustment” that needs to be applied to these propagation regions is that the susceptibility should be reversed in sign there for the Raman-resonant or Kerr-induced FWM or for the SFG and DFG processes, so that these nonlinear processes cannot establish a decrease of the idler intensity in these areas whereas the fields' dephasing can still evolve back to zero. This sign reversal is the ideal case; if this is not possible, then quasi-phase-matching can also be obtained by making the susceptibility zero in the “adjustment” regions. Both the latter type of “adjustment” and the ideal “adjustment” of susceptibility sign reversal is traditionally implemented using a conversion medium where the material properties within the medium are periodically manipulated. This is a complex approach and disadvantageous from a practical point of view. Furthermore, like the dispersion-engineered phase matching approach described earlier, this approach typically yields too large device footprints to realize cost-effective photonic integrated circuits.